I know that 'Nazi' may be overused, but you surely must see that in this specific instance, that is what the Golden Dawn are.

You did mention "using words precisely" at some point. If you still wish to do that, the word "Nazi" is a very specific word -- it refers to things associated with the National Socialist German Workers' Party. I am pretty sure members of Golden Dawn are not members of NSDAP as well.

A better word for you might be "fascist". It is more general -- there certainly were more fascists than nazis -- and describes a particular type of ideology (which originated in Italy, by the way).

The word "neo-nazi", in contemporary parlance, doesn't mean much beyond "I don't like these people".

Note that it's perfectly possible to be racist, xenophobic, nationalist, and anti-semitic and still not be a Nazi or a fascist.